HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.7% → 97.8%

Time: 14.6s

Alternatives: 2

Speedup: 3.1×

Specification

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / sqrt(Float32(Float32(eta * eta) - Float32(Float32(sinTheta_O * sinTheta_O) / sqrt(Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O))))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / sqrt(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrt((single(1.0) - (sinTheta_O * sinTheta_O))))))));
end

Initial Program: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / sqrt(Float32(Float32(eta * eta) - Float32(Float32(sinTheta_O * sinTheta_O) / sqrt(Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O))))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / sqrt(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrt((single(1.0) - (sinTheta_O * sinTheta_O))))))));
end

Alternative 1: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\frac{\frac{eta}{sinTheta_O}}{sinTheta_O}}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (+ eta (/ -0.5 (/ (/ eta sinTheta_O) sinTheta_O))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f / ((eta / sinTheta_O) / sinTheta_O)))));
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / (eta + ((-0.5e0) / ((eta / sintheta_o) / sintheta_o)))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) / Float32(Float32(eta / sinTheta_O) / sinTheta_O)))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) / ((eta / sinTheta_O) / sinTheta_O)))));
end

Derivation

1. Initial program 93.7%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.9%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}}\right) \]
4. Step-by-step derivation

   1. clear-num 97.9%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \color{blue}{\frac{1}{\frac{eta}{{sinTheta_O}^{2}}}}}\right) \]

   2. un-div-inv 97.9%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\frac{-0.5}{\frac{eta}{{sinTheta_O}^{2}}}}}\right) \]

5. Applied egg-rr 97.9%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\frac{-0.5}{\frac{eta}{{sinTheta_O}^{2}}}}}\right) \]
6. Step-by-step derivation

   1. *-un-lft-identity 97.9%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\frac{\color{blue}{1 \cdot eta}}{{sinTheta_O}^{2}}}}\right) \]

   2. unpow2 97.9%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\frac{1 \cdot eta}{\color{blue}{sinTheta_O \cdot sinTheta_O}}}}\right) \]

   3. times-frac 98.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\color{blue}{\frac{1}{sinTheta_O} \cdot \frac{eta}{sinTheta_O}}}}\right) \]

7. Applied egg-rr 98.4%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\color{blue}{\frac{1}{sinTheta_O} \cdot \frac{eta}{sinTheta_O}}}}\right) \]
8. Step-by-step derivation

   1. associate-*l/ 98.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\color{blue}{\frac{1 \cdot \frac{eta}{sinTheta_O}}{sinTheta_O}}}}\right) \]

   2. *-un-lft-identity 98.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\frac{\color{blue}{\frac{eta}{sinTheta_O}}}{sinTheta_O}}}\right) \]

9. Applied egg-rr 98.4%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\color{blue}{\frac{\frac{eta}{sinTheta_O}}{sinTheta_O}}}}\right) \]

10. Final simplification 98.4%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\frac{\frac{eta}{sinTheta_O}}{sinTheta_O}}}\right) \]
11. Add Preprocessing

Alternative 2: 95.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta) :precision binary32 (asin (/ h eta)))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / eta));
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / eta))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / eta))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / eta));
end

Derivation

1. Initial program 93.7%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in eta around inf 95.8%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta}}\right) \]

4. Final simplification 95.8%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta}\right) \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (sinTheta_O h eta)
  :name "HairBSDF, gamma for a refracted ray"
  :precision binary32
  :pre (and (and (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0)) (and (<= -1.0 h) (<= h 1.0))) (and (<= 0.0 eta) (<= eta 10.0)))
  (asin (/ h (sqrt (- (* eta eta) (/ (* sinTheta_O sinTheta_O) (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))